I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

Here are two kinds of spirals for you to explore. What do you notice?

Are these statements always true, sometimes true or never true?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

These tasks give learners chance to generalise, which involves identifying an underlying structure.

Are these statements relating to odd and even numbers always true, sometimes true or never true?

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Make some loops out of regular hexagons. What rules can you discover?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Can you find the values at the vertices when you know the values on the edges?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

How many moves does it take to swap over some red and blue frogs? Do you have a method?

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

It starts quite simple but great opportunities for number discoveries and patterns!

Are these statements always true, sometimes true or never true?

This article for primary teachers discusses how we can help learners generalise and prove, using NRICH tasks as examples.

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Got It game for an adult and child. How can you play so that you know you will always win?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

This activity involves rounding four-digit numbers to the nearest thousand.

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

This challenge asks you to imagine a snake coiling on itself.

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Can you figure out how sequences of beach huts are generated?

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

Can all unit fractions be written as the sum of two unit fractions?